LIPIcs.DISC.2018.9.pdf
- Filesize: 0.5 MB
- 16 pages
Document
Almost Global Problems in the LOCAL Model
Authors
-
Part of:
Volume:
32nd International Symposium on Distributed Computing (DISC 2018)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Symposium on Distributed Computing (DISC) - License:
Creative Commons Attribution 3.0 Unported license
- Publication Date: 2018-10-04
Cite As Get BibTex
Alkida Balliu, Sebastian Brandt, Dennis Olivetti, and Jukka Suomela. Almost Global Problems in the LOCAL Model. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.DISC.2018.9
Abstract
The landscape of the distributed time complexity is nowadays well-understood for subpolynomial complexities. When we look at deterministic algorithms in the LOCAL model and locally checkable problems (LCLs) in bounded-degree graphs, the following picture emerges:
- There are lots of problems with time complexities Theta(log^* n) or Theta(log n).
- It is not possible to have a problem with complexity between omega(log^* n) and o(log n).
- In general graphs, we can construct LCL problems with infinitely many complexities between omega(log n) and n^{o(1)}.
- In trees, problems with such complexities do not exist.
However, the high end of the complexity spectrum was left open by prior work. In general graphs there are problems with complexities of the form Theta(n^alpha) for any rational 0 < alpha <=1/2, while for trees only complexities of the form Theta(n^{1/k}) are known. No LCL problem with complexity between omega(sqrt{n}) and o(n) is known, and neither are there results that would show that such problems do not exist. We show that:
- In general graphs, we can construct LCL problems with infinitely many complexities between omega(sqrt{n}) and o(n).
- In trees, problems with such complexities do not exist.
Put otherwise, we show that any LCL with a complexity o(n) can be solved in time O(sqrt{n}) in trees, while the same is not true in general graphs.
Subject Classification
ACM Subject Classification
- Theory of computation → Distributed computing models
- Theory of computation → Complexity classes
Keywords
- Distributed complexity theory
- locally checkable labellings
- LOCAL model
Metrics
- Access Statistics
-
Total Accesses (updated on a weekly basis)
0PDF Downloads
References
- Alkida Balliu, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Dennis Olivetti, and Jukka Suomela. New classes of distributed time complexity. In Proc. 50th Annual Symposium on the Theory of Computing (STOC 2018). ACM, 2018 (to appear). URL: http://arxiv.org/abs/1711.01871.
- Leonid Barenboim. Deterministic (Δ + 1)-coloring in sublinear (in Δ) time in static, dynamic, and faulty networks. Journal of the ACM, 63(5):47:1-47:22, 2016. URL: http://dx.doi.org/10.1145/2979675.
- Leonid Barenboim, Michael Elkin, and Fabian Kuhn. Distributed (Δ+1)-coloring in linear (in Δ) time. SIAM Journal on Computing, 43(1):72-95, 2014. URL: http://dx.doi.org/10.1137/12088848X.
- Sebastian Brandt, Orr Fischer, Juho Hirvonen, Barbara Keller, Tuomo Lempiäinen, Joel Rybicki, Jukka Suomela, and Jara Uitto. A lower bound for the distributed Lovász local lemma. In Proc. 48th Annual Symposium on the Theory of Computing (STOC 2016), pages 479-488. ACM, 2016. URL: http://dx.doi.org/10.1145/2897518.2897570.
- Sebastian Brandt, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Patric R.J. Östergård, Christopher Purcell, Joel Rybicki, Jukka Suomela, and Przemysław Uznański. LCL problems on grids. In Proc. 35th ACM Symposium on the Principles of Distributed Computing (PODC 2017), pages 101-110, 2017. URL: http://dx.doi.org/10.1145/3087801.3087833.
-
Yi-Jun Chang, Qizheng He, Wenzheng Li, Seth Pettie, and Jara Uitto. The complexity of distributed edge colouring with small palettes. In Proc. 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018). Society for Industrial and Applied Mathematics, 2018.
- Yi-Jun Chang, Tsvi Kopelowitz, and Seth Pettie. An exponential separation between randomized and deterministic complexity in the LOCAL model. In Proc. 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016), pages 615-624. IEEE, 2016. URL: http://arxiv.org/abs/1602.08166.
- Yi-Jun Chang and Seth Pettie. A time hierarchy theorem for the LOCAL model. In Proc. 58th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2017), 2017. URL: http://arxiv.org/abs/1704.06297.
- Richard Cole and Uzi Vishkin. Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control, 70(1):32-53, 1986. URL: http://dx.doi.org/10.1016/S0019-9958(86)80023-7.
- Pierre Fraigniaud, Marc Heinrich, and Adrian Kosowski. Local conflict coloring. In Proc. 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016), pages 625-634, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.73.
- Mohsen Ghaffari and Hsin-Hao Su. Distributed degree splitting, edge coloring, and orientations. In Proc. 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pages 2505-2523. Society for Industrial and Applied Mathematics, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.166.
- Juris Hartmanis and Richard Edwin Stearns. On the computational complexity of algorithms. Transactions of the American Mathematical Society, 117:285-306, 1965. URL: http://dx.doi.org/10.1090/S0002-9947-1965-0170805-7.
-
John E. Hopcroft and Jeffrey D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley, 1979.
- Nathan Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1):193-201, 1992. URL: http://dx.doi.org/10.1137/0221015.
- Moni Naor and Larry Stockmeyer. What can be computed locally? SIAM Journal on Computing, 24(6):1259-1277, 1995. URL: http://dx.doi.org/10.1137/S0097539793254571.
- Alessandro Panconesi and Romeo Rizzi. Some simple distributed algorithms for sparse networks. Distributed Computing, 14(2):97-100, 2001. URL: http://dx.doi.org/10.1007/PL00008932.
- Alessandro Panconesi and Aravind Srinivasan. The local nature of Δ-coloring and its algorithmic applications. Combinatorica, 15(2):255-280, 1995. URL: http://dx.doi.org/10.1007/BF01200759.
-
David Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia, 2000.